Simplify the complex fraction and show all the steps that you used to solve this problem:
3x-7/x^2/ x^2/2+2/x
So it's kinda confusing but the x^2/2+2/x go under the first x^2
The answer I got was
2(3x-7) / x(x^3+4) I just don't know how to explain it
Ok firstly, I'm not very good at utilizing parentheses but I understand how that could've been more helpful. Secondly, I used mathway to type in the equation to get my answer it just wouldn't show me the steps into getting that answer, so I couldn't understand it. That's why I needed some help. Thirdly, thank you for ur help oobleck even though u were a bit rude.
To simplify the given complex fraction, we can follow these steps:
Step 1: Multiply the numerator and the denominator by the least common denominator (LCD) of all the fractions in the complex fraction. In this case, the LCD is x(x^2/2 + 2/x).
To do this, we multiply the numerator by x(x^2/2 + 2/x) and the denominator by x(x^2/2 + 2/x):
[(3x-7) * x(x^2/2 + 2/x)] / [x^2/ x(x^2/2 + 2/x)]
Step 2: Simplify the numerator.
Expanding the numerator: (3x-7) * x(x^2/2 + 2/x) = 3x(x^2/2) + 3x(2/x) - 7x(x^2/2) - 7x(2/x)
Simplifying the numerator: (3x^3/2) + (6x/x) - (7x^3/2) - (14/x)
Combining like terms: (3x^3/2 - 7x^3/2) + (6x - 14)/x = -4x^3/2 + (6x - 14)/x
Step 3: Simplify the denominator.
Expanding the denominator: x^2/ x(x^2/2 + 2/x) = x^2/(x^3/2 + 2)
No further simplification can be done on the denominator.
Step 4: Write the final simplified form.
[(3x-7) * x(x^2/2 + 2/x)] / [x^2/ x(x^2/2 + 2/x)] = (-4x^3/2 + (6x - 14)/x) / (x^2/(x^3/2 + 2))
Now, to make the expression easier to read, we can multiply the numerator by the reciprocal of the denominator (invert and multiply):
[-4x^3/2 + (6x - 14)/x] * [(x^3/2 + 2)/x^2]
= [-4x^3(x^3/2 + 2)/2 + (6x - 14)(x^3/2 + 2)/x] / x^2
Simplifying the numerator further:
= [-2x^6 - 4x^3 + (3x^4 - 7x^3 + 6x - 14)(x^3/2 + 2)/x] / x^2
Combining the terms with x^6 and x^4:
= [-2x^6 - 4x^3 + (3x^7/2 + 6x^4 + 6x^4 - 7x^3x + 6x^3 - 14x^3/2 - 28)/x] / x^2
Simplifying the numerator even further:
= [-2x^6 - 4x^3 + (3x^7/2 + 12x^4 - (7/2)x^4 - (7/2)x^3 + 6x^3 - 14x^3/2 - 28)/x] / x^2
Combining like terms:
= [-2x^6 - 4x^3 + (3x^7/2 + (5/2)x^4 - (9/2)x^3 - 14x^3/2 - 28)/x] / x^2
Finally, we can write the simplified complex fraction as:
(-2x^6 - 4x^3 + 3x^7/2 + (5/2)x^4 - (9/2)x^3 - 14x^3/2 - 28)/x^3
If you got an answer, but can't explain it, did you
(a) just guess?
(b) cheat?
and, did you ever hear of parentheses?
My best guess is
[(3x-7)/x^2] / (x^2/(2+2/x))
= (3x-7)/x^2 * (2+2/x)/(x^2)
= (3x-7)(2+2/x)
= (3x-7) * (2x+2)/x
= (6x^2-8x-14)/x
= 6x - 8 - 14/x
If that is not what you had in mind, then
(a) mosey on over to wolframalpha.com and type in your expression. Use enough parentheses until it interprets your input correctly.
(b) use parens in the future when typing complicated stuff online